The sequence of fractional parts of roots

Kevin O'Bryant. "The sequence of fractional parts of roots." Acta Arithmetica 169.4 (2015): 357-371. <http://eudml.org/doc/286528>.

@article{KevinOBryant2015,
abstract = {We study the function $M_\{θ\}(n) = ⌊1/\{θ^\{1/n\}\}⌋$, where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of $M_\{θ\}$, that if log θ is rational, then for all but finitely many positive integers n, $M_\{θ\}(n) = ⌊n/log θ - 1/2⌋$. We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy $M_\{θ\}(n) = ⌊n/log θ - 1/2⌋$. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued fractions, we produce uncountably many θ that have only finitely many exceptional n, and also give uncountably many explicit θ that have infinitely many exceptional n.},
author = {Kevin O'Bryant},
journal = {Acta Arithmetica},
keywords = {fractional parts of roots; uniform distribution; continued fractions},
language = {eng},
number = {4},
pages = {357-371},
title = {The sequence of fractional parts of roots},
url = {http://eudml.org/doc/286528},
volume = {169},
year = {2015},
}

TY - JOUR
AU - Kevin O'Bryant
TI - The sequence of fractional parts of roots
JO - Acta Arithmetica
PY - 2015
VL - 169
IS - 4
SP - 357
EP - 371
AB - We study the function $M_{θ}(n) = ⌊1/{θ^{1/n}}⌋$, where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of $M_{θ}$, that if log θ is rational, then for all but finitely many positive integers n, $M_{θ}(n) = ⌊n/log θ - 1/2⌋$. We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy $M_{θ}(n) = ⌊n/log θ - 1/2⌋$. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued fractions, we produce uncountably many θ that have only finitely many exceptional n, and also give uncountably many explicit θ that have infinitely many exceptional n.
LA - eng
KW - fractional parts of roots; uniform distribution; continued fractions
UR - http://eudml.org/doc/286528
ER -